Cartesian Product is Small
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Theorem
Let $a$ and $b$ be small classes.
Then their Cartesian product $\left({a \times b}\right)$ is small:
- $\mathscr M \left({a \times b}\right)$
Proof
| \(\displaystyle \left({a \times b}\right)\) | \(=\) | \(\displaystyle \left\{ {\left({x, y}\right) : x \in a \land y \in b}\right\}\) | Definition of Cartesian Product | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left\{ {\left({x, y}\right) : \left\{ {x}\right\} \subseteq a \land \left\{ {y}\right\} \subseteq b} \right\}\) | Singleton of Element is Subset | ||||||||||
| \(\displaystyle \) | \(\subseteq\) | \(\displaystyle \left\{ {\left({x, y}\right) : \left\{ {x, y}\right\} \subseteq \left({a \cup b}\right)}\right\}\) | Set Union Preserves Subsets | ||||||||||
| \(\displaystyle \) | \(\subseteq\) | \(\displaystyle \left\{ {\left({x, y}\right) : \left({x, y}\right) \subseteq \mathcal P \left({a \cup b}\right)}\right\}\) | Power Set of Subset and Subset Relation is Transitive |
It's not obvious how $\left({x, y}\right) \subseteq \mathcal P \left({a \cup b}\right)$
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
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So by the definition of power set:
- $\left({a \times b}\right) \subseteq \mathcal P \left({\mathcal P \left({a \cup b}\right)}\right)$
By Union of Small Classes is Small, $\left({a \cup b}\right)$ is small.
By the Axiom of Powers, $\mathcal P \left({\mathcal P \left({a \cup b}\right)}\right)$ is small.
By Axiom of Subsets Equivalents, $\left({a \times b}\right)$ is small.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.2$
